how to copy a rig to another mesh
Duplicating an armature with ShiftD only works while in object mode and it will only create a duplicate armature in the same file. Also CtrlC only copies the armature object in object mode, in pose mode it will copy the bone locations allowing you to copy the pose.

In your case, with two models in a separate files, you could also open the second blend file then Select Append from the file menu. Link will use the same armature while append will copy the armature into the current file.



When you select a blend file it will show the following list -



First go into object and select the armature you want. Then append again and go into Action and select the actions you want to import into the new file.

As for shape keys, they are part of the mesh object data. If the two mesh objects have similar topology you may be able to copy the shape keys between models. You could append the first mesh object and then try to copy the shapekeys then delete the first object.

I think you may find it easier to append the second object to the same file as the first one, duplicate the armature, copy shapekeys etc. Then save a duplicate copy of the blend and remove one model from each file.